Spectral radius of weighted composition operators in $L^p$-spaces

Krzysztof Zajkowski ¹

¹ Institute of Mathematics University of Białystok Akademicka 2 15-267 Białystok, Poland

Studia Mathematica, Tome 198 (2010) no. 3, pp. 301-307

Résumé

We prove that for the spectral radius of a weighted composition operator $aT_\alpha$, acting in the space $L^p(X,\mathcal{B},\mu)$, the following variational principle holds: $$ \ln r(aT_\alpha)=\max_{\nu\in M^1_{\alpha,{\rm e}}}\int_X\ln|a|\,d\nu, $$ where $X$ is a Hausdorff compact space, $\alpha:X\to X$ is a continuous mapping preserving a Borel measure $\mu$ with $\mathop{\rm supp}\mu=X$, $M^1_{\alpha,{\rm e}}$ is the set of all $\alpha$-invariant ergodic probability measures on~$X$, and $a:X\to \mathbb{R}$ is a continuous and $\mathcal{B}_\infty$-measurable function, where $\mathcal{B}_\infty=\bigcap_{n=0}^\infty\alpha^{-n}(\mathcal{B})$. This considerably extends the range of validity of the above formula, which was previously known in the case when $\alpha$ is a homeomorphism.

Détail

DOI : 10.4064/sm198-3-8

Keywords: prove spectral radius weighted composition operator alpha acting space mathcal following variational principle holds alpha max alpha int where hausdorff compact space alpha continuous mapping preserving borel measure mathop supp alpha set alpha invariant ergodic probability measures mathbb continuous mathcal infty measurable function where mathcal infty bigcap infty alpha n mathcal considerably extends range validity above formula which previously known alpha homeomorphism